Polymer-chain adsorption transition at a cylindrical boundary

In a recent paper [S. Boettcher and M. Moshe, Phys. Rev. Lett. 74, 2410 (1995)], a simple method was proposed to generate solvable models that predict the critical properties of statistical systems in hyperspherical geometries. To that end, it was shown how to reduce a random walk in D dimensions to an anisotropic one-dimensional random walk on concentric hyperspheres. Here I construct such a random walk to model the adsorption-desorption transition of polymer chains growing near an attractive cylindrical boundary such as that of a cell membrane. I find that the fraction of adsorbed monomers on the boundary vanishes exponentially when the adsorption energy decreases towards its critical value. When the adsorption energy rises beyond a certain value above the critical point whose scale is set by the radius of the cell, the adsorption fraction exhibits a crossover to a linear increase characteristic of polymers growing near planar boundaries.